FRANCESCO CARAVENNA, RONGFENG SUN, AND NIKOS ZYGOURAS
Abstract. Any renewal processes on N0with a polynomial tail, with exponent α ∈ (0, 1), has a non-trivial scaling limit, known as the α-stable regenerative set. In this paper we consider Gibbs transformations of such renewal processes in an i.i.d. random environment, called disordered pinning models. We show that for α ∈ (12, 1) these models have a universal scaling limit, which we call the continuum disordered pinning model (CDPM). This is a random closed subset of R in a white noise random environment, with subtle features:
• Any fixed a.s. property of the α-stable regenerative set (e.g., its Hausdorff dimension) is also an a.s. property of the CDPM, for almost every realization of the environment.
• Nonetheless, the law of the CDPM is singular with respect to the law of the α-stable regenerative set, for almost every realization of the environment.
The existence of a disordered continuum model, such as the CDPM, is a manifestation of disorder relevance for pinning models with α ∈ (12, 1).
1. Introduction
We consider disordered pinning models, which are defined via a Gibbs change of measure of a renewal process, depending on an external i.i.d. random environment. First introduced in the physics and biology literature, these models have attracted much attention due to their rich structure, which is amenable to a rigorous investigation; see, e.g., the monographs of Giacomin [G07, G10] and den Hollander [dH09].
In this paper we define a continuum disordered pinning model (CDPM), inspired by recent work of Alberts, Khanin and Quastel [AKQ14b] on the directed polymer in random environment. The interest for such a continuum model is manifold:
• It is a universal object, arising as the scaling limit of discrete disordered pinning models in a suitable continuum and weak disorder limit, cf. Theorem 1.3.
• It provides a tool to capture the emerging effect of disorder in pinning models, when disorder is relevant, cf. Subsection 1.4 for a more detailed discussion.
• It can be interpreted as an α-stable regenerative set in a white noise random environ- ment, displaying subtle properties, cf. Theorems 1.4, 1.5 and 1.6.
Throughout the paper, we use the conventions N := {1, 2, . . .}, N0 := {0} ∪ N, and write an∼ bn to mean limn→∞an/bn= 1.
1.1. Renewal processes and regenerative sets. Let τ := (τn)n≥0be a renewal process on N0, that is τ0= 0 and the increments (τn− τn−1)n∈N are i.i.d. N-valued random
Date: December 12, 2014.
1991 Mathematics Subject Classification. Primary: 82B44; Secondary: 82D60, 60K35.
Key words and phrases. Scaling Limit, Disorder Relevance, Weak Disorder, Pinning Model, Fell-Matheron Topology, Hausdorff Metric, Random Polymer, Wiener Chaos Expansion.
1
variables (so that 0 = τ0 < τ1 < τ2 < . . .). Probability and expectation for τ will be denoted respectively by P and E. We assume that τ is non-terminating, i.e., P(τ1 < ∞) = 1, and
K(n) := P(τ1= n) = L(n)
n1+α, as n → ∞ , (1.1)
where α ∈ (0, 1) and L(·) is a slowly varying function [BGT87]. We assume for simplicity that K(n) > 0 for every n ∈ N (periodicity complicates notation, but can be easily incorporated).
Let us denote by C the space of all closed subsets of R. There is a natural topology on C, called the Fell-Matheron topology [F62, M75, M05], which turns C into a compact Polish space, i.e. a compact separable topological space which admits a complete metric. This can be taken as a version of the Hausdorff distance (see Appendix A for more details).
Identifying the renewal process τ = {τn}n≥0 with its range, we may view τ as a random subset of N0, i.e. as a C-value random variable (hence we write {n ∈ τ } :=S
k≥0{τk= n}).
This viewpoint is very fruitful, because as N → ∞ the rescaled set τ
N = τn N
n≥0
(1.2) converges in distribution on C to a universal random closed set τ of [0, ∞), called the α-stable regenerative set (cf. [FFM85], [G07, Thm. A.8]). This coincides with the closure of the range of the α-stable subordinator or, equivalently, with the zero level set of a Bessel process of dimension δ = 2(1 − α) (see Appendix A), and we denote its law by Pα.
Remark 1.1. Random sets have been studied extensively [M75, M05]. Here we focus on the special case of random closed subsets of R. The theory developed in [FFM85] for regenerative sets cannot be applied in our context, because we modify renewal processes through inhomogeneous perturbations and conditioning (see (1.4)-(1.9) below). For this reason, in Appendix A we review and develop a general framework to study convergence of random closed sets of R, based on a natural notion of finite-dimensional distributions.
1.2. Disordered pinning models. Let ω := (ωn)n∈N be i.i.d. random variables (independent of the renewal process τ ), which represent the disorder. Probability and
expectation for ω will be denoted respectively by P and E. We assume that
E[ωn] = 0, Var(ωn) = 1, ∃t0> 0 : Λ(t) := log E[etωn] < ∞ for |t| ≤ t0. (1.3) The disordered pinning model is a random probability law PωN,β,h on subsets of {0, . . . , N }, indexed by realizations ω of the disorder, the system size N ∈ N, the disorder strength β > 0 and bias h ∈ R, defined by the following Gibbs change of measure of the renewal process τ :
PωN,β,h(τ ∩ [0, N ])
P(τ ∩ [0, N ]) := 1
ZN,β,hω ePNn=1(βωn−Λ(β)+h)1{n∈τ }, (1.4) where the normalizing constant
ZN,β,hω := E h
ePNn=1(βωn−Λ(β)+h)1{n∈τ } i
(1.5) is called the partition function. In words, we perturb the law of the renewal process τ in the interval [0, N ], by giving rewards/penalties (βωn− Λ(β) + h) to each visited site n ∈ τ . (The presence of the factor Λ(β) in (1.4)-(1.5), which just corresponds to a translation of h,
allows to have normalized weights E[eβωn−Λ(β)] = 1 for h = 0.)
The properties of the model PωN,β,h, especially in the limit N → ∞, have been studied in depth in the recent mathematical literature (see e.g. [G07, G10, dH09] for an overview). In this paper we focus on the problem of defining a continuum analogue of PωN,β,h.
Since under the “free law” P the rescaled renewal process τ /N converges in distribution to the α-stable regenerative set τ , it is natural to ask what happens under the “interacting law” PωN,β,h. Heuristically, in the scaling limit the i.i.d. random variables (ωn)n∈N should be replaced by a one-dimensional white noise (dWt)t∈[0,∞), where W = (Wt)t∈[0,∞) denotes a standard Brownian motion (independent of τ ). Looking at (1.4), a natural candidate for the scaling limit of τ /N under PωN,β,h would be the random measure Pα;WT ,β,hon C defined by
dPα;WT ,β,h
dPα (τ ∩ [0, T ]) := 1 Zα;WT,β,he
RT
0 1{t∈τ }(βdWt+(h−12β2)dt), (1.6) where the continuum partition function Zα;WT ,β,hwould be defined in analogy to (1.5). The problem is that a.e. realization of the α-stable regenerative set τ has zero Lebesgue measure, hence the integral in (1.6) vanishes, yielding the “trivial” definition Pα;WT,β,h= Pα.
These difficulties turn out to be substantial and not just technical: as we shall see, a non- trivial scaling limit of PωN,β,h does exist, but, for α ∈ (12, 1), it is not absolutely continuous with respect to the law Pα (hence no formula like (1.6) can hold). Note that an analogous phenomenon happens for the directed polymer in random environment, cf. [AKQ14b].
1.3. Main results. We need to formulate an additional assumption on the renewal processes that we consider. Introducing the renewal function
u(n) := P(n ∈ τ ) =
∞
X
k=0
P(τk= n),
assumption (1.1) yields u(n + `)/u(n) → 1 as n → ∞, provided ` = o(n) (see (2.10) below).
We ask that the rate of this convergence is at least a power-law of n`:
∃C, n0 ∈ (0, ∞), ε, δ ∈ (0, 1] :
u(n + `) u(n) −1
≤ C ` n
δ
∀n ≥ n0, 0 ≤ ` ≤ εn . (1.7) Remark 1.2. As we discuss in Appendix B, condition (1.7) is a very mild smoothness requirement, that can be verified in most situations. E.g., it was shown by Alexander [A11]
that for any α > 0 and slowly varying L(·), there exists a Markov chain X on N0 with ±1 steps, called Bessel-like random walk, whose return time to 0, denoted by T , is such that
K(n) := P(T = 2n) = L(n)e
n1+α as n → ∞, with eL(n) ∼ L(n). (1.8) We prove in Appendix B that any such walk always satisfies (1.7).
Recall that C denotes the compact Polish space of closed subsets of R. We denote by M1(C) the space of Borel probability measures on C, which is itself a compact Polish space, equipped with the topology of weak convergence. We will work with a conditioned version of the disordered pinning model (1.4), defined by
Pω,cN,β,h( · ) := PωN,β,h( · |N ∈ τ ) . (1.9) (In order to lighten notation, when N 6∈ N we agree that Pω,cN,β,h:= Pω,cbN c,β,h.)
Recalling (1.2), let us introduce the notation Pω,cN T,β
N,hN(d(τ /N )) := law of the rescaled set τ
N ∩ [0, T ] under Pω,cN T ,β
N,hN. (1.10) For a fixed realization of the disorder ω, Pω,cN T,β
N,hN(d(τ /N )) is a probability law on C, i.e.
an element of M1(C). Since ω is chosen randomly, the law Pω,cN T,β
N,hN(d(τ /N )) is a random element of M1(C), i.e. a M1(C)-valued random variable.
Our first main result is the convergence in distribution of this random variable, provided α ∈ (12, 1) and the coupling constants β = βN and h = hN are rescaled appropriately:
βN := ˆβL(N )
Nα−12 , hN := ˆhL(N )
Nα , for N ∈ N, ˆβ > 0, ˆh ∈ R . (1.11) Theorem 1.3 (Existence and universality of the CDPM). Fix α ∈ (12, 1), T > 0, β > 0, ˆˆ h ∈ R. There exists a M1(C)-valued random variable Pα;W,c
T, ˆβ,ˆh, called the (conditioned) continuum disordered pinning model (CDPM), which is a function of the parameters (α, T, ˆβ, ˆh) and of a standard Brownian motion W = (Wt)t≥0, with the following property:
• for any renewal process τ satisfying (1.1) and (1.7), and βN, hN defined as in (1.11);
• for any i.i.d. sequence ω satisfying (1.3);
the law Pω,cN T,β
N,hN(d(τ /N )) of the rescaled pinning model, cf. (1.10), viewed as a M1(C)- valued random variable, converges in distribution to Pα;W,c
T, ˆβ,ˆh as N → ∞.
We refer to Subsection 1.4 for a discussion on the universality of the CDPM. We stress that the restriction α ∈ (12, 1) is substantial and not technical, being linked with the issue of disorder relevance, as we explain in Subsection 1.4 (see also [CSZ13]).
Let us give a quick explanation of the choice of scalings (1.11). This is the canonical scaling under which the partition function ZN,βω
N,hN in (1.5) has a nontrivial continuum limit. To see this, write
ZN,β,hω = E hYN
n=1
(1 + εβ,hn 1n∈τ) i
= 1 +
N
X
k=1
X
1≤n1<···<nk≤N
εβ,hn1 · · · εβ,hn
k P(n1 ∈ τ, ..., nk ∈ τ ), where εβ,hn := eβωn−Λ(β)+h− 1. By Taylor expansion, as β, h tend to zero, one has the asymptotic behavior E[εβ,hn ] ≈ h and Var(εβ,hn ) ≈ β2. Using this fact, we see that the asymptotic mean and variance behavior of the first term (k = 1) in the above series is
E hXN
n=1
εβ,hn P(n ∈ τ ) i
≈ h
N
X
n=1
P(n ∈ τ ) ≈ h Nα L(N ),
Var hXN
n=1
εβ,hn P(n ∈ τ )i
≈ β2
N
X
n=1
P(n ∈ τ )2 ≈ β2N2α−1 L(N )2 ,
because P(n ∈ τ ) ≈ nα−1/L(n), by (1.1) (see (2.10) below). Therefore, for these quantities to have a non-trivial limit as N tends to infinity, we are forced to scale βN and hN as in (1.11). Remarkably, this is also the correct scaling for higher order terms in the expansion
for ZN,β,hω , as well as for the measure PωN,β
N,hN to converge to a non-trivial limit.
We now describe the continuum measure. For a fixed realization of the Brownian motion W = (Wt)t∈[0,∞), which represents the “continuum disorder”, we call Pα;W,c
T , ˆβ,ˆh the quenched law of the CDPM, while
E h
Pα;W,c
T , ˆβ,ˆh
i ( · ) :=
Z
Pα;W,c
T, ˆβ,ˆh( · ) P(dW ) (1.12) will be called the averaged law of the CDPM. We also introduce, for T > 0, the law Pα;cT of the α-stable regenerative set τ restricted on [0, T ] and conditioned to visit T :
Pα;cT ( · ) := Pα(τ ∩ [0, T ] ∈ · | T ∈ τ ) , (1.13) which will be called the reference law. (Relation (1.13) is defined through regular conditional distributions.) Note that both EPα;W,c
T , ˆβ,ˆh and Pα;cT are probability laws on C, while Pα;W,c
T, ˆβ,ˆh
is a random probability law on C.
Intuitively, the quenched law Pα;W,c
T , ˆβ,ˆh could be conceived as a “Gibbs transformation” of the reference law Pα;cT , where each visited site t ∈ τ ∩ [0, T ] of the α-stable regenerative set is given a reward/penalty ˆβdWdtt + ˆh, like in the discrete case. This heuristic interpretation should be taken with care, however, as the following results show.
Theorem 1.4 (Absolute Continuity of the Averaged CDPM). For all α ∈ (12, 1), T > 0, ˆβ > 0, ˆh ∈ R, the averaged law EPα;W,c
T , ˆβ,ˆh of the CDPM is absolutely continuous with respect to the reference law Pα;cT . It follows that any typical property of the reference law Pα;cT is also a typical property of the quenched law Pα;W,c
T , ˆβ,ˆh, for a.e. realization of W :
∀A ⊆ C such that Pα;cT (A) = 1 : Pα;W,c
T , ˆβ,ˆh(A) = 1 for P-a.e. W . (1.14) In particular, for a.e. realization of W , the quenched law Pα;W,c
T , ˆβ,ˆh of the CDPM is supported on closed subsets of [0, T ] with Hausdorff dimension α.
It is tempting to deduce from (1.14) the absolute continuity of the quenched law Pα;W,c
T, ˆβ,ˆh
with respect to the reference law Pα;cT , for a.e. realization of W , but this is false.
Theorem 1.5 (Singularity of the Quenched CDPM). For all α ∈ (12, 1), T > 0, ˆβ > 0, ˆh ∈ R and for a.e. realization of W , the quenched law Pα;W,cT , ˆβ,ˆh of the CDPM is singular with respect to the reference law Pα;cT :
for P-a.e. W , ∃A ⊆ C such that Pα;cT (A) = 1 and Pα;W,c
T , ˆβ,ˆh(A) = 0 . (1.15) The seeming contradiction between (1.14) and (1.15) is resolved noting that in (1.14) one cannot exchange “∀A ⊆ C” and “for P-a.e. W ”, because there are uncountably many A ⊆ C (and, of course, the set A appearing in (1.15) depends on the realization of W ).
We conclude our main results with an explicit characterization of the CDPM. As we discuss in Appendix A, each closed subset C ⊆ R can be identified with two non-decreasing and right-continuous functions gt(C) and dt(C), defined for t ∈ R by
gt(C) := sup{x : x ∈ C ∩ [−∞, t]}, dt(C) := inf{x : x ∈ C ∩ (t, ∞]} . (1.16)
As a consequence, the law of a random closed subset X ⊆ R is uniquely determined by the finite dimensional distributions of the random functions (gt(X))t∈Rand (dt(X))t∈R, i.e. by the probability laws on R2k given, for k ∈ N and −∞ < t1 < t2 < . . . < tk< ∞, by
P gt1(X) ∈ dx1, dt1(X) ∈ dy1, . . . , gtk(X) ∈ dxk, dtk(X) ∈ dyk . (1.17) As a further simplification, it is enough to focus on the event that X ∩ [ti, ti+1] 6= ∅ for all i = 1, . . . , k, that is, one can restrict (x1, y1, . . . , xk, yk) in (1.17) on the following set:
R(k)t
0,...,tk+1 :=(x1, y1, . . . , xk, yk) : xi∈ [ti−1, ti], yi∈ [ti, ti+1] for i = 1, . . . , k,
such that yi ≤ xi+1 for i = 1, . . . , k − 1 , (1.18) with t0 = −∞ and tk+1 := +∞. The measures (1.17) restricted on the set (1.18) will be called restricted finite-dimensional distributions (f.d.d.) of the random set X (see §A.3).
We can characterize the CDPM by specifying its restricted f.d.d.. We need two ingredients:
(1) The restricted f.d.d. of the α-stable regenerative set conditioned to visit T , i.e. of the reference law Pα;cT in (1.13): by Proposition A.8, these are absolutely continuous with respect to the Lebesgue measure on R2k, with the following density (with y0 := 0):
fT ;tα;c
1,...,tk(x1, y1, . . . , xk, yk) =
k
Y
i=1
Cα
(xi− yi−1)1−α(yi− xi)1+α
! T1−α
(T − yk)1−α , (1.19) with Cα:=α sin(πα)
π , (1.20)
where we restrict (x1, y1, . . . , xk, yk) on the set (1.18), with t0 = 0 and tk+1 := T . (2) A family of continuum partition functions for our model:
Zα;W,cˆ
β,ˆh (s, t)
0≤s≤t<∞.
These were constructed in [CSZ13] as the limit, in the sense of finite-dimensional distributions, of the following discrete family (under an appropriate rescaling):
Zβ,hω,c(a, b) := E h
ePb−1n=a+1(βωn−Λ(β)+h)1{n∈τ }
a ∈ τ, b ∈ τ i
, 0 ≤ a ≤ b . (1.21) In Section 2 we upgrade the f.d.d. convergence to the process level, deducing important a.s. properties, such as strict positivity and continuity (cf. Theorems 2.1 and 2.4).
We can finally characterize the restricted f.d.d. of the CDPM as follows.
Theorem 1.6 (F.d.d. of the CDPM). Fix α ∈ (12, 1), T > 0, ˆβ > 0, ˆh ∈ R and let Zα;W,cˆ
β,ˆh (s, t)
0≤s≤t<∞ be an a.s. continuous version of the continuum partition functions.
For a.e. realization of W , the quenched law Pα;W,c
T, ˆβ,ˆh of the CDPM (cf. Theorem 1.3) can be defined as the unique probability law on C which satisfies the following properties:
(i) Pα;W,c
T, ˆβ,ˆh is supported on closed subsets τ ⊆ [0, T ] with {0, T } ⊆ τ .
(ii) For all k ∈ N and 0 =: t0 < t1 < · · · < tk < tk+1 := T , and for (x1, y1, . . . , xk, yk) restricted on the set R(k)t
0,...,tk+1 in (1.18), the f.d.d. of Pα;W,c
T , ˆβ,ˆh have densities given by Pα;W,c
T , ˆβ,ˆh gt1(τ ) ∈ dx1, dt1(τ ) ∈ dy1, . . . , gtk(τ ) ∈ dxk, dtk(τ ) ∈ dyk dx1dy1 · · · dxkdyk
= Qk
i=0Zα;W,cˆ
β,ˆh (yi, xi+1) Zα;W,cˆ
β,ˆh (0, T )
! fT ;tα;c
1,...,tk(x1, y1, . . . , xk, yk) ,
(1.22)
where we set y0 := 0 and xk+1 := T , and where fT ;tα;c
1,...,tk(·) is defined in (1.19).
1.4. Discussion and perspectives. We conclude the introduction with some obser- vations on the results stated so far, putting them in the context of the existing literature, stating some conjectures and outlining further directions of research.
1. (Disorder relevance). The parameter β tunes the strength of the disorder in the model Pω,cN,β,h, cf. (1.9), (1.4). When β = 0, the sequence ω disappears and we obtain the so-called homogeneous pinning model. Roughly speaking, the effect of disorder is said to be:
• irrelevant if the disordered model (β > 0) has the same qualitative behavior as the homogeneous model (β = 0), provided the disorder is sufficiently weak (β 1);
• relevant if, on the other hand, an arbitrarily small amount of disorder (any β > 0) alters the qualitative behavior of the homogeneous model (β = 0).
Recalling that α is the exponent appearing in (1.1), it is known that disorder is irrelevant for pinning models when α < 12 and relevant when α > 12, while the case α = 12 is called marginal and is more delicate (see [G10] and the references therein for an overview).
It is natural to interpret our results from this perspective. For simplicity, in the sequel we set hN := ˆh L(N )/Nα, as in (1.11), and we use the notation Pω,cN T,β
N,hN(d(τ /N )), cf. (1.10), for the law of the rescaled set τ /N under the pinning model.
In the homogeneous case (β = 0), it was shown in [Soh09, Theorem 3.1]†that the weak limit of Pα;cN T,0,h
N(d(τ /N )) as N → ∞ is a probability law Pα;c
T ,0,ˆh on C which is absolutely continuous with respect to the reference law Pα;cT (recall (1.13)):
dPα;c
T,0,ˆh
dPα;cT (τ ) = eˆhLT(τ )
E[eˆhLT(τ )], (1.23)
where LT(τ ) denotes the so-called local time associated to the regenerative set τ . We stress that this result holds with no restriction on α ∈ (0, 1).
Turning to the disordered model β > 0, what happens for α ∈ (0,12)? In analogy with [B89, CY06], we conjecture that for fixed β > 0 small enough, the limit in distribution of Pω,cN T,β,h
N(d(τ /N )) as N → ∞ is the same as for the homogeneous model (β = 0), i.e. the law Pα;c
T ,0,ˆh defined in (1.23). Thus, for α ∈ (0,12), the continuum model is non-disordered (deterministic) and absolutely continuous with respect to the reference law.
†Actually [Soh09] considers the non-conditioned case (1.4), but it can be adapted to the conditioned case.
This is in striking contrast with the case α ∈ (12, 1), where our results show that the continuum model Pα;W,c
T, ˆβ,ˆh is truly disordered and singular with respect to the reference law (cf. Theorems 1.3, 1.4, 1.5). In other terms, for α ∈ (12, 1), disorder survives in the scaling limit (even though βN, hN → 0) and breaks down the absolute continuity with respect to the reference law, providing a clear manifestation of disorder relevance.
We refer to [CSZ13] for a general discussion on disorder relevance in our framework.
2. (Universality). The quenched law Pα;W,c
T , ˆβ,ˆh of the CDPM is a random probability law on C, i.e. a random variable taking values in M1(C). Its distribution is a probability law on the space M1(C) —i.e. an element of M1(M1(C))— which is universal : it depends on few macroscopic parameters (the time horizon T , the disorder strength and bias ˆβ, ˆh and the exponent α) but not on finer details of the discrete model from which it arises, such as the distributions of ω1 and of τ1: all these details disappear in the scaling limit.
Another important universal aspect of the CDPM is linked to phase transitions. We do not explore this issue here, referring to [CSZ13, §1.3] for a detailed discussion, but we mention that the CDPM leads to sharp predictions about the asymptotic behavior of the free energy and critical curve of discrete pinning models, in the weak disorder regime λ, h → 0.
3. (Bessel processes). In this paper we consider pinning models built on top of general renewal processeses τ = (τk)k∈N0 satisfying (1.1) and (1.7). In the special case when the renewal process is the zero level set of a Bessel-like random walk [A08] (recall Remark 1.2), one can define the pinning model (1.4), (1.9) as a probability law on random walk paths (and not only on their zero level set).
Rescaling the paths diffusively, one has an analogue of Theorem 1.3, in which the CDPM is built as a random probability law on the space C([0, T ], R) of continuous functions from [0, T ] to R. Such an extended CDPM is a continuous process (Xt)t∈[0,T ], that can be heuristically described as a Bessel process of dimension δ = 2(1 − α) interacting with an independent Brownian environment W each time Xt = 0. The “original” CDPM of our Theorem 1.3 corresponds to the zero level set τ := {t ∈ [0, T ] : Xt= 0}.
We stress that, starting from the zero level set τ , one can reconstruct the whole process (Xt)t∈[0,T ] by pasting independent Bessel excusions on top of τ (more precisely, since the open set [0, T ] \ τ is a countable union of disjoint open intervals, one attaches a Bessel excursion to each of these intervals).†This provides a rigorous definition of (Xt)t∈[0,T ] in terms of τ and shows that the zero level set is indeed the fundamental object.
4. (Infinite-volume limit). Our continuum model Pα;W,c
T , ˆβ,ˆh is built on a finite interval [0, T ].
An interesting open problem is to let T → ∞, proving that Pα;W,c
T, ˆβ,ˆh converges in distribution to an infinite-volume CDPM Pα;W,c
∞, ˆβ,ˆh. Such a limit law would inherit scaling properties from the continuum partition functions, cf. Theorem 2.4 (iii). (See also [RVY08] for related work in the non-disordered case ˆβ = 0.)
1.5. Organization of the paper. The rest of the paper is organized as follows.
• In Section 2, we study the properties of continuum partition functions.
†Alternatively, one can write down explicitly the f.d.d. of (Xt)t∈[0,T ]in terms of the continuum partition functions Zα;W,cˆ
β,ˆh (s, t) (see Section 2). We skip the details for the sake of brevity.
• In Section 3, we prove Theorem 1.6 on the characterization of the CDPM, which also yields Theorem 1.3.
• In Section 4, we prove Theorems 1.4 and 1.5 on the relations between the CDPM and the α-stable regenerative set.
• In Appendix A, we describe the measure-theoretic background needed to study random closed subsets of R, which is of independent interest.
• Lastly, in Appendices B and C we prove some auxiliary estimates.
2. Continuum partition functions as a process In this section we focus on a family Zα;W,cˆ
β,ˆh (s, t)
0≤s≤t<∞of continuum partition functions for our model, which was recently introduced in [CSZ13] as the limit of the discrete family (1.21) in the sense of finite-dimensional distributions. We upgrade this convergence to the process level (Theorem 2.1), which allows us to deduce important properties (Theorem 2.4).
Besides their own interest, these results are the key to the construction of the CDPM.
2.1. Fine properties of continuum partition functions. Recalling (1.21), where Zβ,hω,c(a, b) is defined for a, b ∈ N0, we extend Zβ,hω,c(·, ·) to a continuous function on
[0, ∞)2≤ := {(s, t) ∈ [0, ∞)2 : 0 ≤ s ≤ t < ∞} ,
bisecting each unit square [m − 1, m] × [n − 1, n], with m ≤ n ∈ N, along the main diagonal and linearly interpolating Zβ,hω,c(·, ·) on each triangle. In this way, we can regard
Zβω,c
N,hN(sN, tN )
0≤s≤t<∞ (2.1)
as random variables taking values in the space C([0, ∞)2≤, R), equipped with the topology of uniform convergence on compact sets and with the corresponding Borel σ-algebra. The randomness comes from the disorder sequence ω = (ωn)n∈N.
Even though our main interest in this paper is for α ∈ (12, 1), we also include the case α > 1 in the following key result, which is proved in Subsection 2.2 below.
Theorem 2.1 (Process Level Convergence of Partition Functions). Let α ∈ (12, 1) ∪ (1, ∞), ˆβ > 0, ˆh ∈ R. Let τ be a renewal process satisfying (1.1) and (1.7), and ω be an
i.i.d. sequence satisfying (1.3). For every N ∈ N, define βN, hN by (recall (1.11))
βN := ˆβL(N ) Nα−12 hN := ˆhL(N ) Nα
for α ∈ (12, 1) ,
βN := βˆ
√ N hN := ˆh
N
for α > 1 . (2.2)
As N → ∞ the two-parameter family Zβω,c
N,hN(sN, tN )
0≤s≤t<∞ converges in distribution on C([0, ∞)2≤, R) to a family Zα;W,cβ,ˆˆh (s, t)
0≤s≤t<∞, called continuum partition functions.
For all 0 ≤ s ≤ t < ∞, one has the Wiener chaos representation Zα;W,cˆ
β,ˆh (s, t) = 1 +
∞
X
k=1
Z
· · · Z
s<t1<···<tk<t
ψα;cs,t(t1, . . . , tk)
k
Y
i=1
( ˆβ dWti+ ˆh dti) , (2.3)
where W = (Wt)t≥0 is a standard Brownian motion, the series in (2.3) converges in L2, and the kernel ψα;cs,t(t1, . . . , tk) is defined as follows, with Cα as in (1.20) and t0:= s:
ψα;cs,t(t1, . . . , tk) =
k
Y
i=1
Cα (ti− ti−1)1−α
!(t − s)1−α
(t − tk)1−α if α ∈ (12, 1), 1
E[τ1]k if α > 1.
(2.4)
Remark 2.2. The integral in (2.3) is defined by expanding formally the product of differ- entials and reducing to standard multiple Wiener and Lebesgue integrals. An alternative equivalent definition is to note that, by Girsanov’s theorem, the law of ( ˆβWt+ ˆht)t∈[0,T ] is absolutely continuous w.r.t. that of ( ˆβWt)t∈[0,T ], with Radon-Nikodym density
fT , ˆβ,ˆh(W ) := e(
ˆh
βˆ) WT−12(ˆhˆ
β)2T
. (2.5)
It follows that Zα;W,cˆ
β,ˆh (s, t)
0≤s≤t≤T has the same law as Zα;W,cˆ
β,0 (s, t)
0≤s≤t≤T (for ˆh = 0) under a change of measure with density (2.5). For further details, see [CSZ13].
Remark 2.3. Theorem 2.1 still holds if we also include the two-parameter family of unconditioned partition functions Zβω
N,hN(sN, tN )
0≤s≤t<∞, defined the same way as Zβ,hω,c(a, b) in (1.21), except for removing the conditioning on b ∈ τ . The limiting process Zα;Wˆ
β,ˆh (s, t) will then have a kernel ψαs,t, which modifies ψα;cs,t in (2.4), by setting ψαs,t(t1, . . . , tk) =
k
Y
i=1
Cα
(ti− ti−1)1−α, if α ∈ (12, 1). (2.6) By Theorem 2.1, we can fix a version of the continuum partition functions Zα;W,cˆ
β,ˆh (s, t) which is continuous in (s, t). This will be implicitly done henceforth. We can then state some fundamental properties, proved in Subsection 2.3.
Theorem 2.4 (Properties of Continuum Partition Functions). For all α ∈ (12, 1), β > 0, ˆˆ h ∈ R the following properties hold:
(i) (Positivity) For a.e. realization of W , the function (s, t) 7→ Zα;W,cˆ
β,ˆh (s, t) is continuous and strictly positive at all 0 ≤ s ≤ t < ∞.
(ii) (Translation Invariance) For any fixed t > 0, the process (Zα;W,cˆ
β,ˆh (t, t + u))u≥0 has the same distribution as (Zα;W,cˆ
β,ˆh (0, u))u≥0, and is independent of (Zα;W,cˆ
β,ˆh (s, u))0≤s≤u≤t. (iii) (Scaling Property) For any constant A > 0, one has the equality in distribution
Zα;W,cˆ
β,ˆh (As, At)
0≤s≤t<∞
dist=
Zα;W,c
Aα−1/2β,Aˆ αˆh(s, t)
0≤s≤t<∞. (2.7) (iv) (Renewal Property) Setting Z(s, t) := Zα;W,cˆ
β,ˆh (s, t) for simplicity, for a.e. realization of W one has, for all 0 ≤ s < u < t < ∞,
CαZ(s, t) (t − s)1−α =
Z
x∈(s,u)
Z
y∈(u,t)
CαZ(s, x) (x − s)1−α
1 (y − x)1+α
CαZ(y, t)
(t − y)1−α dx dy , (2.8)
which can be rewritten, recalling (1.19), as follows:
Z(s, t) = Eα;ct−s h
Z s, gu(τ ) Z du(τ ), ti
. (2.9)
The rest of this section is devoted to the proof of Theorems 2.1 and 2.4. We recall that assumption (1.1) entails the following key renewal estimates, with Cα as in (1.20):
u(n) := P(n ∈ τ ) ∼
Cα
L(n)n1−α if 0 < α < 1, 1
E[τ1]= (const.) ∈ (0, ∞) if α > 1,
(2.10)
by the classical renewal theorem for α > 1 and by [GL63, D97] for α ∈ (0, 1). Let us also note that the additional assumption (1.7) for α ∈ (0, 1) can be rephrased as follows:
|u(q) − u(r)| ≤ Cr − q r
δ
u(q) , ∀r ≥ q ≥ n0 with r − q ≤ εr , (2.11) up to a possible change of the constants C, n0, ε.
2.2. Proof of Theorem 2.1. We may assume T = 1. For convergence in distribution on C([0, 1]2≤, R) it suffices to show that {(Zβω,cN,hN(sN, tN ))0≤s≤t≤1}N ∈N is a tight family, because the finite-dimensional distribution convergence was already obtained in [CSZ13]
(see Theorem 3.1 and Remark 3.3 therein). We break down the proof into five steps.
Step 1. Moment criterion. We recall a moment criterion for the Hölder continuity of a family of multi-dimensional stochastic processes, which was also used in [AKQ14a] to prove similar tightness results for the directed polymer model. Using Garsia’s inequality [G72, Lemma 2] with Ψ(x) = |x|p and ϕ(u) = uq for p ≥ 1 and pq > 2d, the modulus of continuity of a continuous function f : [0, 1]d→ R can be controlled by
|f (x) − f (y)| ≤ 8 Z |x−y|
0
Ψ−1
B u2d
dϕ(u) = 8 Z |x−y|
0
B1/p
u2d/pd(uq) = 8B1/pq
q − 2d/p|x − y|q−2d/p, where
B = B(f ) = Z Z
[0,1]d×[0,1]d
Ψ
f (x) − f (y) ϕ x−y√
d
dx dy = dq/2 Z Z
[0,1]d×[0,1]d
|f (x) − f (y)|p
|x − y|pq dx dy.
Suppose now that (fN)N ∈N are random continuous function on [0, 1]d such that E[|fN(x) − fN(y)|p] ≤ C|x − y|η,
for some C, p, q, η ∈ (0, ∞) with pq > 2d and η > pq − d, uniformly in N ∈ N, x, y ∈ [0, 1]d. Then E[B(fN)] is bounded uniformly in N , hence {B(fN)}N ∈N is tight. If the functions fN are equibounded at some point (e.g. fN(0) = 1 for every N ∈ N), the tightness of B(fN) entails the tightness of {fN}N ∈N, by the Arzelà-Ascoli theorem [B99, Theorem 7.3].
To prove the tightness of {(Zβω,c
N,hN(sN, tN ))0≤s≤t≤1}N ∈N, it then suffices to show that E
h Zβω,c
N,hN(s1N, t1N ) − Zβω,c
N,hN(s2N, t2N )
pi
≤ C p
(s1− s2)2+ (t1− t2)2η
, (2.12) which by triangle inequality, translation invariance and symmetry can be reduced to
∃C > 0, p ≥ 1, η > 2 : E h
Zβω,c
N,hN(0, tN ) − Zβω,c
N,hN(0, sN )
pi
≤ C|t − s|η, (2.13)
uniformly in N ∈ N and 0 ≤ s < t ≤ 1. (Conditions pq > 2d and η > pq − d are then fulfilled by any q ∈ (4p,2+ηp ), since d = 2). Since Zβω,c
N,hN(0, ·) is defined on [0, ∞) via linear interpolation, it suffices to prove (2.13) for s, t with sN, tN ∈ {0} ∪ N.
Step 2. Polynomial chaos expansion. To simplify notation, let us denote
ΨN,r:= Zβω,c
N,hN(0, r) = E
"r−1 Y
n=1
e(βNωn−Λ(βN)+hN)1{n∈τ }
r ∈ τ
#
for r ∈ N,
and ΨN,0:= 1. Since ex1{n∈τ } = 1 + (ex− 1)1{n∈τ } for all x ∈ R, we set
ξN,i:= eβNωi−Λ(βN)+hN − 1, (2.14) and rewrite ΨN,r as a polynomial chaos expansion:
ΨN,r= E
"r−1 Y
i=1
(1 + ξN,i1{i∈τ })
r ∈ τ
#
= X
I⊂{1,...,r−1}
P(I ⊂ τ |r ∈ τ )Y
i∈I
ξN,i, (2.15)
using the notation {I ⊂ τ } :=T
i∈I{i ∈ τ }.
Recalling (2.2) and (1.3), it is easy to check that E[ξN,i] = ehN− 1 = hN + O(h2N),
q
Var(ξN,i) = q
e2hN eΛ(2βN)−2Λ(βN)− 1 =q
βN2 + O(βN3) = βN+ O(βN2),
(2.16)
where we used the fact that hN = o(βN) and we Taylor expanded Λ(t) := log E[etω1], noting that Λ(0) = Λ0(0) = 0 and Λ00(0) = 1. Thus hN and βN are approximately the mean and standard deviation of ξN,i. Let us rewrite ΨN,r in (2.15) using normalized variables ζN,i:
ΨN,r= X
I⊂{1,...,r−1}
ψN,r(I)Y
i∈I
ζN,i, where ζN,i:= 1
βNξN,i, (2.17) where ψN,r(∅) := 1 and for I = {n1< n2< · · · < nk} ⊂ N, recalling (2.10), we can write
ψN,r(I) = ψN,r(n1, . . . , nk) := βN|I|P(I ⊂ τ |r ∈ τ ) = (βN)k 1 u(r)
k+1
Y
i=1
u(ni− ni−1), (2.18) with n0 := 0, nk+1:= r.
To prove (2.13), we write Zβω,c
N,hN(0, sN ) = ΨN,q and Zβω,c
N,hN(0, tN ) = ΨN,r, with q := sN and r := tN , so that 0 ≤ q < r ≤ N . For a given truncation level m = m(q, r, N ) ∈ (0, q), that we will later choose as
m = m(q, r, N ) :=
(0 if q ≤pN (r − q)
q −pN (r − q) otherwise , (2.19)
so that 0 ≤ m < q < r ≤ N , we write
ΨN,r− ΨN,q = Ξ1+ Ξ2− Ξ3
with
Ξ1 = X
I⊂{1,...,m}
ψN,r(I) − ψN,q(I) Y
i∈I
ζN,i,
Ξ2 = X
I⊂{1,...,r−1}
I∩{m+1,...,r−1}6=∅
ψN,r(I)Y
i∈I
ζN,i, and Ξ3 = X
I⊂{1,...,q−1}
I∩{m+1,...,q−1}6=∅
ψN,q(I)Y
i∈I
ζN,i. (2.20) To establish (2.13) and hence tightness, it suffices to show that for each i = 1, 2, 3,
∃C > 0, p ≥ 1, η > 2 : E[|Ξi|p] ≤ Cr − q N
η
∀N ∈ N, 0 ≤ q < r ≤ N. (2.21) Step 3. Change of measure. We now estimate the moments of ξN,i defined in (2.14).
Since (a + b)2k≤ 22k−1(a2k+ b2k), for all k ∈ N, and hN = O(βN2 ) by (2.2), we can write E[ξN,i2k] ≤ 22k−1e2k(hN−Λ(βN))E
eβNωi − 12k + 22k−1(e−Λ(βN)+hN − 1)2k
≤ C(k) βN2kE h 1
βN
Z βN
0
ωietωidt
2ki
+ O(βN4k+ h2kN)
≤ C(k)βN2k−1 Z βN
0
E[ωi2ke2ktωi]dt + o(βN2k) = O(βN2k),
(2.22)
because E[ω2ki e2ktωi] is uniformly bounded for t ∈ [0, t0/4k] by our assumption (1.3).
Recalling (2.16), (2.17) and (2.2), the random variables (ζN,i)i∈N are i.i.d. with E[ζN,i] ∼
N →∞
ˆh βˆ
√1
N , Var[ζN,i] ∼
N →∞ 1, sup
N,i∈NE[(ζN,i)2k] < ∞. (2.23) It follows, in particular, that {ζN,i2 }i,N ∈Nare uniformly integrable. We can then apply a change of measure result established in [CSZ13, Lemma B.1], which asserts that we can construct i.i.d.
random variables (eζN,i)i∈N with marginal distribution P(eζN,i∈ dx) = fN(x)P(ζN,i∈ dx), for which there exists C > 0 such that for all p ∈ R and i, N ∈ N
E[eζN,i] = 0, E[eζN,i2 ] ≤ 1 + C/
√
N , and E[fN(ζN,i)p] ≤ 1 + C/N. (2.24) Let eΞi be the analogue of Ξi constructed from the eζN,i’s instead of the ζN,i’s. By Hölder,
E|Ξi|l−1 = Eh
|Ξi|l−1
N
Y
i=1
fN(ζN,i)l−1l
N
Y
i=1
fN(ζN,i)−l−1l i
≤ E|eΞi|ll−1l
EfN(ζN,1)1−lNl
≤ eClE|eΞi|ll−1l . Relation (2.21), and hence the tightness of {Zβω,c
N,hN(·, ·)}N ∈N, is thus reduced to showing E|eΞi|l ≤ Cr − q
N
η
for all N ∈ N and 0 ≤ q < r ≤ N, (2.25) for some l ∈ N, l ≥ 2 and η > 0 satisfying η > 2l−1l .
Step 4. Bounding E[|eΞ2|l]. We note that the bound for E[|eΞ3|l] is exactly the same as that for E[|eΞ2|l], and hence will be omitted. First we write eΞ2 as
Ξe2 =
r−1
X
k=1
Ξe(k)2 , where Ξe(k)2 := X
|I|=k,I⊂{1,...,r−1}
I∩{m+1,...,r−1}6=∅
ψN,r(I)Y
i∈I
ζeN,i, (2.26)
with eΞ(k)2 consisting of all terms of degree k. The hypercontractivity established in [MOO10, Prop. 3.16 & 3.12] allows to estimates moments of order l in terms of moments of order 2:
more precisely, setting kXkp := E[|X|p]1/p, we have for all l ≥ 2
keΞ2kll := E[|eΞ2|l] ≤
r−1X
k=1
keΞ(k)2 kll
≤Xr−1
k=1
(cl)kkeΞ(k)2 k2l
, (2.27)
where cl:= 2√
l − 1 maxN ∈Nkeζ
N,1kl
keζN,1k2
is finite and depends only on l, by (2.23).
We now turn to the estimation of keΞ(k)2 k2. Let us recall the definition of ψN,r in (2.18). It follows by (2.24) that Var(eζN,1) ≤ 1 + C/√
N ≤ 2 for all N large. We then have
keΞ(k)2 k22= E Ξe(k)2 2
=
k−1
X
y=0
X
1≤n1<···<ny ≤m m+1≤ny+1<···<nk≤r−1
ψN,r2 (n1, . . . , nk) Var(eζN,1)k
≤ 2k
k−1
X
y=0
X
1≤n1<···<ny ≤m m+1≤ny+1<···<nk≤r−1
βN2ku(n1)2u(n2− n1)2· · · u(r − nk)2 u(r)2
≤ 4k
k−1
X
y=0
Z
· · · Z
0<t1<···<ty <m m N
N<ty+1<···<tk<r N
(√
N βNu(dN t1e))2· · · (√
N βNu(r − dN tke))2 (√
N βNu(r))2 dt1· · · dtk. (2.28)
It remains to estimate this integral, when 12 < α < 1 (the case α > 1 is easy). By (2.10) 1
c
1
L(` + 1)(` + 1)1−α ≤ u(`) ≤ c 1
L(` + 1)(` + 1)1−α ∀` ∈ N , for some c ∈ (0, ∞). Since dN te − dN se + 1 ≥ N (t − s), recalling (2.2) we obtain
√
N βNu dN te − dN se ≤ c L(N )
L dN te − dN se + 1 1 (t − s)1−α . Let us now fix
α0 :=
(1 when α > 1
any number in 12, α
when 12 < α < 1. (2.29) Since L(·) is slowly varying, by Potter bounds [BGT87, Theorem 1.5.6] for every ε > 0 there is Dε∈ (0, ∞) such that L(a)/L(b) ≤ Dεmax{(a/b)ε, (b/a)ε} for all a, b ∈ N. It follows that
√
N βNu dN te − dN se ≤ C 1
(t − s)1−α0, (2.30)
for some C ∈ (0, ∞), uniformly in 0 < s < t ≤ 1 and N ∈ N. Analogously, again by (2.10) and Potter bounds, if 0 ≤ s < t < Nr we have
u(dN te − dN se)
u(r) ≤ c2 L(r + 1) L(dN te − dN se + 1)
r1−α
(N (t − s))1−α ≤ C(r/N )1−α0
(t − s)1−α0. (2.31)
